Oriented diameter of graphs with given maximum degree

Oriented diameter of graphs with given maximum degree In this article, we show that every bridgeless graph G of order n and maximum degree Δ has an orientation of diameter at most n−Δ+3. We then use this result and the definition NG(H)=⋃v∈V(H)NG(v)∖V(H), for every subgraph H of G, to give better bounds in the case that G contains certain clusters of high‐degree vertices, namely: For every edge e, G has an orientation of diameter at most n−|NG(e)|+4, if e is on a triangle and at most n−|NG(e)|+5, otherwise. Furthermore, for every bridgeless subgraph H of G, there is such an orientation of diameter at most n−|NG(H)|+3. Finally, if G is bipartite, then we show the existence of an orientation of diameter at most 2(|A|− deg G(s))+7, for every partite set A of G and s∈V(G)∖A. This particularly implies that balanced bipartite graphs have an orientation of diameter at most n−2Δ+7. For each bound, we give a polynomial‐time algorithm to construct a corresponding orientation and an infinite family of graphs for which the bound is sharp. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Graph Theory Wiley

Oriented diameter of graphs with given maximum degree

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Publisher
Wiley Subscription Services, Inc., A Wiley Company
Copyright
Copyright © 2018 Wiley Periodicals, Inc.
ISSN
0364-9024
eISSN
1097-0118
D.O.I.
10.1002/jgt.22181
Publisher site
See Article on Publisher Site

Abstract

In this article, we show that every bridgeless graph G of order n and maximum degree Δ has an orientation of diameter at most n−Δ+3. We then use this result and the definition NG(H)=⋃v∈V(H)NG(v)∖V(H), for every subgraph H of G, to give better bounds in the case that G contains certain clusters of high‐degree vertices, namely: For every edge e, G has an orientation of diameter at most n−|NG(e)|+4, if e is on a triangle and at most n−|NG(e)|+5, otherwise. Furthermore, for every bridgeless subgraph H of G, there is such an orientation of diameter at most n−|NG(H)|+3. Finally, if G is bipartite, then we show the existence of an orientation of diameter at most 2(|A|− deg G(s))+7, for every partite set A of G and s∈V(G)∖A. This particularly implies that balanced bipartite graphs have an orientation of diameter at most n−2Δ+7. For each bound, we give a polynomial‐time algorithm to construct a corresponding orientation and an infinite family of graphs for which the bound is sharp.

Journal

Journal of Graph TheoryWiley

Published: Jan 1, 2018

Keywords: ; ; ; ; ;

References

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